## T-test vs F-test vs Z-test | Difference between T-test, F-test, Z-test

This page compares T-test vs F-test vs Z-test and mentions **difference between** T-test, F-test and Z-test.
The applications of T-test, F-test and Z-test are also mentioned.

**What is hypothesis**:

• It is a predictive statement capable of being tested by scientific methods.

• Hypothesis is simple proposition which can be proved or disproved through
various scientific techniques.

• It establishes relationship between independent variable to some dependent variable.

• All these three are hypothesis tests.

• Entire decision to validate or to refuse the null hypothesis is based on
these tests. Hence it is essential to choose the right test.

### T-test

• It is a univariate hypothesis test.

• It is applied in condition when standard deviation is not known and
sample size is small (n < 30).

• It is used to compare means of two populations.

• T-test is commonly used compare to z-test as explained below.

• Test statistic and assumption for T-test is mentioned for one sample test.
Similarly two sample t-test, paired t-test are performed for
various applications.

### F-test

• It is a statistical test.

• It is used to determine equality of variances of two normal populations.

• It is used to compare two population variances.

• This test is based on snedecor f-distribution (under null hypothesis).

• Test statistic for F-test is mentioned. It is known as one-way ANOVA test.
It is used to test difference of three or more population means.

### Z-test

• It is also statistical hypothesis test which follows normal
distribution.

• It is applied to moderate to large samples (i.e. when n > 30).

• It is used to determine whether two population means are different when
variances are known and sample size is large as mentioned.

• Z-test depends on certain conditions in order to be reliable.
Hence it is less adaptable compare to T-test.

• Z-test is preferable compare to T-test when standard deviation values are available or
known.

• Test statistic and assumption for Z-test is mentioned for one sample test.

Following table summarizes difference between these parametric test types.

Features |
F-test |
T-test |
Z-test |
---|---|---|---|

Application | Comparing variances of two samples | Comparing the mean to a value, or the means of two samples | same as T-test but for large samples |

Used when | 3 or more μ_{s} |
n < 30 and σ is unknown | n >= 30 and σ is known |

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